# Regular embedding

In algebraic geometry, a closed immersion of schemes is a **regular embedding** of codimension *r* if each point *x* in *X* has an open affine neighborhood *U* in *Y* such that the ideal of is generated by a regular sequence of length *r*.

For example, if *X* and *Y* are smooth over a scheme *S* and if *i* is an *S*-morphism, then *i* is a regular embedding. In particular, every section of a smooth morphism is a regular embedding.^{[1]} If is regularly embedded into a regular scheme, then *B* is a complete intersection ring.^{[2]}

The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when *i* is a regular embedding, if *I* is the ideal sheaf of *X* in *Y*, then the normal sheaf, the dual of , is locally free (thus a vector bundle) and the natural map is an isomorphism: the normal cone coincides with the normal bundle.

A flat morphism of finite type is called a **(local) complete intersection morphism** if each point *x* in *X* has an open affine neighborhood *U* so that *f* |_{U} factors as where *j* is a regular embedding and *g* is smooth.^{[3]} For example, if *f* is a morphism between smooth varieties, then *f* factors as where the first map is the graph morphism and so is a complete intersection morphism.

## References

- Fulton, William (1998),
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics],**2**, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7 - E. Sernesi:
*Deformations of algebraic schemes*